Complete Question
The diagram of this question is shown on the first uploaded image
Answer:
The distance the block slides before stopping is ![d = 0.313 \ m](https://tex.z-dn.net/?f=d%20%3D%200.313%20%5C%20m)
Explanation:
The free body diagram for the diagram in the question is shown
From the diagram the angle is ![\theta = 25 ^o](https://tex.z-dn.net/?f=%5Ctheta%20%3D%2025%20%5Eo)
![sin \theta = \frac{h}{d}](https://tex.z-dn.net/?f=sin%20%5Ctheta%20%20%3D%20%5Cfrac%7Bh%7D%7Bd%7D)
Where ![h = h_b - h_a](https://tex.z-dn.net/?f=h%20%3D%20h_b%20-%20h_a)
So ![d sin \theta = h_b - h_a](https://tex.z-dn.net/?f=d%20sin%20%5Ctheta%20%20%3D%20h_b%20-%20h_a)
From the question we are told that
The mass of the block is ![m = 20 \ kg](https://tex.z-dn.net/?f=m%20%3D%2020%20%5C%20kg)
The mass of the pendulum is ![m_p = 2 \ kg](https://tex.z-dn.net/?f=m_p%20%3D%202%20%5C%20kg)
The velocity of the pendulum at the bottom of swing is ![v_p = 15 m/s](https://tex.z-dn.net/?f=v_p%20%3D%2015%20m%2Fs)
The coefficient of restitution is ![e =0.7](https://tex.z-dn.net/?f=e%20%3D0.7)
The coefficient of kinetic friction is ![\mu _k = 0.5](https://tex.z-dn.net/?f=%5Cmu%20_k%20%3D%200.5)
The velocity of the block after the impact is mathematically represented as
![v_2 f = \frac{m_b - em_p}{m_b + m_p} * v_2 i + \frac{[1 + e] m_1}{m_1 + m_2 } v_p](https://tex.z-dn.net/?f=v_2%20f%20%3D%20%5Cfrac%7Bm_b%20-%20em_p%7D%7Bm_b%20%2B%20m_p%7D%20%20%2A%20v_2%20i%20%2B%20%5Cfrac%7B%5B1%20%2B%20e%5D%20m_1%7D%7Bm_1%20%2B%20m_2%20%7D%20v_p)
Where
is the velocity of the block before collision which is 0
![= \frac{20 - (0.7 * 2)}{(2 + 20)} * 0 + \frac{(1 + 0.7) * 2 }{2 + 20} * 15](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B20%20-%20%280.7%20%2A%202%29%7D%7B%282%20%2B%2020%29%7D%20%2A%200%20%2B%20%5Cfrac%7B%281%20%2B%200.7%29%20%2A%202%20%7D%7B2%20%2B%2020%7D%20%20%20%2A%2015)
Substituting value
![v_2 f = 2.310\ m/s](https://tex.z-dn.net/?f=v_2%20f%20%3D%202.310%5C%20%20m%2Fs)
According to conservation of energy principle
The energy at point a = energy at point b
So ![PE_A + KE _A = PE_B + KE_B + E_F](https://tex.z-dn.net/?f=PE_A%20%2B%20KE%20_A%20%3D%20PE_B%20%2B%20KE_B%20%20%2B%20%20E_F)
Where
is the potential energy at A which is mathematically represented as
= 0 at the bottom
is the kinetic energy at A which is mathematically represented as
is the potential energy at B which is mathematically represented as
From the diagram ![h = h_b -h_a](https://tex.z-dn.net/?f=h%20%3D%20h_b%20-h_a)
![PE_B = m_b g(h_b - h_a)](https://tex.z-dn.net/?f=PE_B%20%3D%20m_b%20g%28h_b%20-%20h_a%29)
is the kinetic energy at B which is 0 (at the top )
Where is
is the workdone against velocity which from the diagram is
![\mu_k m_b g cos 25 *d](https://tex.z-dn.net/?f=%5Cmu_k%20m_b%20g%20cos%2025%20%2Ad)
So
![\frac{1}{2} m_b v_2 f^2 = m_b g h_b + \mu_k m_b g cos \25 * d](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20m_b%20v_2%20f%5E2%20%20%3D%20m_b%20g%20h_b%20%2B%20%5Cmu_k%20m_b%20g%20cos%20%5C25%20%2A%20d)
Substituting values
So
![d = 0.313 \ m](https://tex.z-dn.net/?f=d%20%3D%200.313%20%5C%20m)